Convergence of a sequence
Given a real number , the sequence is defined recursively by:
(i) Does approach a limit as increases without bound?
(ii) How does this limit, when it exists, depend on ?
(iii) When there is a limit, how rapidly does converge to the limit? Does this depend on ?
This is an investigation in analysis, which is quite hard without recourse to computational methods. It is a nice example of how computational techniques can provide information about a strange sequence.
More generally, what is the behavior of sequences defined recursively by: